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Events
Workshop on geometric infinite surfaces

MPI für Mathematik in den Naturwissenschaften Leipzig

E1 05 (Leibniz-Saal)

conference

16.09.24

Workshop on geometric infinite surfaces

The workshop on Geometric Infinite Surfaces is focused on bringing together mathematicians who work with surfaces, particularly using geometric tools and sometimes even working with surfaces of infinite type. In addition to 4 main speakers, we will have lighting talks and discussion sessions to foster new connections.

List of speakers:

Myfanwy Evans (Potsdam University)
Magali Jay (MPI MiS)
Hana Kouřimská (IST Austria)
Yuriy Tumarkin (University of Zurich)

Program

10:00 - 10:15	Welcome coffee
10:15 - 11:15	Magali Jay (Max Planck Institute for Mathematics in the Sciences) Tiling billiard in the wind-tree model In this talk, I will present the meeting of different dynamical systems: tiling billiards, the wind-tree model and Eaton lenses. The three of them are motivated by physics. In the beginning of the 2000's, physicists have conceived metamaterials with negative index of refraction. Tilling billiards' trajectories consist of light rays moving in a arrangement of metamaterials with opposite indew of refraction. The wind-tree model was instoduced by Paul and Tatyana Ehrenfest to study a gaz: a particule is moving in a plane where obstacles are periodically placed, on which the particule bounces. The Eaton lenses are a periodic array of lenses in the plane, in which we consider a light ray that is reflected each time it crosses a lens. After having introduced these dynamical systems, I will consider a mix of them: an arrangement of rectangles in the plane, like in the wind-tree model, but made of metamaterials, like for tiling billiards. I study the trajectories of light in this plane. They are refracted each time they cross a rectangle. I show that these trajectories are traped in a strip, for almost every parameter. This behavior is similar to the one of Eaton lenses.
11:15 - 11:30	Skye Rothstein (MPI MiS) Folding Cubical Landscapes A polyomino is a discrete 2D geometric object built by arranging a set of congruent squares edge to edge. Popularized by games like Tetris, polyominoes are extensively studied in recreational math, among other fields. We say a polyomino is “fenestrated” if it encloses a hole. We approach the study of fenestrated polyominoes with a background in kirigami (origami with cutting) to explore folding 3D cubical landscapes.
11:30 - 11:45	Felix Günther (Technische Universität Berlin) Discrete Riemann surfaces In this lightning talk, we provide an overview of previous research findings and current research interests in the field of discrete Riemann surfaces. We begin with discrete holomorphic differentials and discrete period matrices, continue with a study of discrete real Riemann surfaces, and conclude with some first insights into discrete spin structures.
11:45 - 12:00	Samantha Fairchild (Max Planck Institute for Mathematics in the Sciences) The Mutetrahedron The mutetrahedron is a triply periodic surface whose quotient by a lattice in 3-space gives a compact half-translation surface admitting many symmetries. We will give a friendly introduction to the muetrahedron and one of its friends called the mucube. We will then highlight some explicit computations focused on understanding the behavior of geodesics on the Mutetrahedron. This is based on an ongoing project with Dami Lee and Sunrose Shrestha.
12:00 - 12:15	James Farre (Max Planck Institute for Mathematics in the Sciences) Horocycles, Lipschitz maps, and laminations I will overview some recent results classifying horocycle orbit closures in Z-covers of closed hyperbolic surfaces and highlight the interplay between the geometry of best Lipschitz maps to the circle, dynamics of horocyles, and topology of a canonical geodesic lamination recovered from both perspectives. This is joint work with Or Landesberg and Yair Minsky.
12:15 - 12:45	Lunch
12:45 - 13:45	Dami Lee (Oklahoma State University) A family of triply periodic polyhedral surfaces Inspired by the rich theory of triply periodic minimal surfaces, we study a family of triply periodic polyhedral surfaces that have several conformal representations. When quotiented by the translation lattice, we get a compact Riemann surface. Some of these examples answer questions in minimal surface theory, Teichmüller dynamics, and physics.
13:45 - 14:45	Yuriy Tumarkin (Universität Zürich) Invariant measures for the linear flow on infinite translation surfaces Translation surfaces are a class of locally flat surfaces that can be obtained by gluing together edges of a collection of polygons. There is a natural flow that can be defined on a translation surface, the straight-line flow in a given direction. The dynamics of this flow is very well understood for compact translation surfaces, with classical results telling us that for almost all directions, Lebesgue is the unique invariant measure for the flow. For translation surfaces of infinite type however the picture is very different, and very little is still known about the classification of invariant measures of the linear flow. I will talk about the simplest case, where the infinite type surface is the Z^d cover of a compact translation surface and further the infinite surface is periodic under a renormalisation map. In this case it turns out that one can classify the ergodic invariant measures and even obtain a somewhat explicit form for them.
14:45 - 15:00	Coffee & Tea
15:00 - 16:00	Hana Dal Poz Kourimska (University of Potsdam) Uniformization theorem for triangulated piecewise linear surfaces Uniformization theorem is one of the pillars of 19th century mathematics. In terms of closed orientable Riemannian 2-manifolds, it states that every such surface is conformally equivalent to a unique closed 2-manifold of constant positive, zero, or negative curvature. I will talk about how the uniformization theorem and the whole differential-geometric machinery behind it has been translated into the realm of discrete surfaces, and adapted to be suitable for computations.
16:00 - 17:30	Open problems discussion

Participants

Toky Andriamanalina

University of Potsdam

Irina Bobrova

MPI MiS

Fernando Camacho Cadena

Max Planck Institute for Mathematics in the Sciences

Türkü Özlüm Çelik

MPI CBG

Hana Dal Poz Kourimska

University of Potsdam

Myfanwy Evans

University of Potsdam

Lander Fabian

MPI MIS

Samantha Fairchild

Max Planck Institute for Mathematics in the Sciences

James Farre

Max Planck Institute for Mathematics in the Sciences

Felix Günther

Technische Universität Berlin

Magali Jay

Max Planck Institute for Mathematics in the Sciences

Kanishka Katipearachchi

Humboldt University, Berlin

Laura Lankers

MPI MiS Leipzig

Dami Lee

Oklahoma State University

Jose Antonio Ramirez Rafael

Leipzig University

Bernhard Reinke

MPI MiS

Max Riestenberg

MPI MIS Leipzig

Skye Rothstein

MPI MiS

Jiajun Shi

MPI MiS

Yuriy Tumarkin

Universität Zürich

Organizers

Samantha Fairchild

Max Planck Institute for Mathematics in the Sciences

James Farre

Max Planck Institute for Mathematics in the Sciences

Administrative Contact

Antje Vandenberg

Max Planck Institute for Mathematics in the Sciences Contact via Mail